Yes, "şu" is pronounced like shoe. But "su" means water which is different. So you have to use "şu" for both cases. ("That bread" and "That is bread") is same in Turkish) But the stressing is slightly different so we can distinguish.

There is. It's "olmak". However, as with other languages, you don't say "I be a Doctor", and olmak becomes modified where it's used, the same way it becomes am or is in English. Except in Turkish, it's used as a suffix and not a separate word.

The upside is, you have the added benefit of being able to drop it in most cases (if not, all), because it's considered implied.

So, for example, you can say "eşim doktor", literally, "my spouse doctor [is]". The "is" is implied, and not used. Another example: "Ben, Tim", literally, "I, Tim [am]". The "am" is implied, and is dropped.

If you wanted to use "to be" in those sentences, it wouldn't be wrong, nor would it sound strange. People often drop it, because it's just easier. You would simply say, "eşim doktordur" and "Ben, Tim'im".

So, going back to the original exercise, if you want to be absolutely clear that your meaning is "that is bread", you can say "şu, ekmektir" (literally, "that, bread is").

There are two kind of pointing sth/sb in English: this(for close ones) and that(for far ones). But in Turkish there are three : bu(for close ones), şu(for the ones a little bit away from you) and o(for farther ones).

It is similar to the ones in Japanese:
koko-kono:bu
soko-sono:şu
ako-ano:o